On two matrix-free continuation approaches for the determination of the bifurcation diagram of the von Kármán system
Keywords:
Mixed finite element, Newton-GMRES, ANM, Padé approximationAbstract
In this paper we present a combination of the asymptotic numerical continuation procedure with a preconditioned GMRES-solver as applied to the fourth-order non linear von Kármán problem. Using a mixed equal order finite element discretization, we show how our “matrix free”approach allows for an efficient determination of the non-trivial branch of the bifurcation diagram. We also show that, using a steplength estimation of Gervais and Sadiky [GER 04], one can limit himself to a third order prediction without loosing too much in the number of continuation steps.
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References
[ALL 90] ALLGOWER E. L., GEORG K., Numerical continuation methods, Springer-Verlag,
Berlin, 1990.
[CIA 80] CIARLET P. G., RABIER P., Les équations de von Kármán, Springer, Berlin, 1980.
[COC 94] COCHELIN B., “A path following technique via an Asymptotic Numerical Method”,
Computers and Structures, vol. 53, num. 5, 1994, p. 1181-1192.
[DOS 03] DOSSOU K., PIERRE R., “A Newton-GMRES approach for the analysis of the postbuckling
behavior of the solutions of the von Kármán equations”, SIAM Journal of scientific
computing, vol. 24, num. 6, 2003, p. 1994-2012.
[GER 97] GERVAIS J.-J., OUKIT A., PIERRE R., “Finite element analysis of the buckling and
mode jumping of a rectangular plate”, Dynam. Stability Systems, vol. 12, num. 3, 1997,
p. 161-185.
[GER 02] GERVAIS J., SADIKY H., “A new steplength control for continuation with the
asymptotic numerical method”, IMA J. Numer. Anal., vol. 22, 2002, p. 207-229.
[GER 04] GERVAIS J., SADIKY H., “A continuation method based on a high order predictor
and an adaptive steplength control”, ZAMM., vol. 84, num. 7, 2004.
[HOL 84] HOLDER E. J., SCHAEFFER D., “Boundary conditions and mode jumping in the
von Kármán equations”, SIAM J. Math. Anal., vol. 15, num. 3, 1984, p. 446-458.
[HUN 80] HUNTER C., GUERRIERI B., “Deducing the properties of singularities of functions
from their Taylor series coefficients”, SIAM J. Appl. Math., vol. 39, 1980, p. 248-263.
[KEL 77] KELLER H., “Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problem”,
RABINOWITZ P. H., Ed., Applications of bifurcation theory, Academic Press, New-
York, 1977.
[SAD 00] SADIKY H., “Une classe de méthodes de continuation utilisant le développement de
Taylor”, PhD thesis, Université Laval, Québec, Canada, 2000.
[ZAH 04] ZAHROUNI H., AGGOUNE W., BRUNELOT J., POTIER-FERRY M., “Asymptotic
numerical method for strong non-linearities”, Revue Européenne des Éléments Finis,
vol. 13, 2004.
